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Tuesday, 24 July 2012

The sum may not converge for all values of ‘z’. The value of ‘z’ for which the sum converges is called Region of Convergence (ROC).

PROPERTIES OF REGION OF CONVERGENCE

1.The ROC is a concentric ring or a circle in the z-plane centered at the origin.

2.The ROC cannot contain any poles.

3.If x(n) is a finite duration causal sequence, the ROC is entire z-plane except at z=0.

If x(n) is a finite duration anti causal sequence, then the ROC is the entire z-plane except at z=∞.
If x(n) is a finite duration 2-sided sequence, then the ROC will the entire z-plane except at z=0 and z=∞.

4.If x(n) is a right sided sequence and if the circle |z|=r0 is in the ROC, then all finite values of ‘z’ for which |z|>ro will also be in ROC.

5. If x(n) is a left sided sequence and if the circle |z|=r0 is in the ROC, then all values of z for which 0<|z|<ro will be in ROC.

6. If x(n) is a 2-sided sequence and if the circle |z|=r0 is in the ROC, then the ROC will consists of a ring in the z-plane that includes the circle |z|=r0

7. If the z-Transform X(z) of x(n) is rational, then its ROC is bounded by poles or extends to ‘∞’.

8. If the z-Transform X(z) of x(n) is rational and if x(n) is right sided, then ROC is the region in the z-plane outside the outermost pole. In other words, Outside the radius of circle = the largest magnitude of pole of x(z). If x(n) is causal then ROC also includes Z= ∞.

9. If the z-Transform X(z) of x(n) is rational and if x(n) is left sided, then ROC is the region in the z-plane inside the outermost ‘non zero pole’.

In other words, inside the circle of radius = the smallest magnitude of pole of x(z) other than at z=0 and extending inwards to and possibly including z=0. If x(n) is anti-causal, ROC includes z=0.

10.If x(n) is a finite duration 2-sided sequence, then ROC will consists of a circular ring in the z-plane bounded on the interior and exterior by a pole and not containing any pole.

11. The ROC of an LTI (Linear Time Invariant) system contains the unit circle.

The Z-Transform is the discrete time counter part of Laplace transform. Z-transform allows us to perform transform analysis of unstable systems and to develop additional insights and tools for LTI (linear Time Invariant)system analysis. The Z-transform transforms difference equation into algebraic equations and hence the discrete time system analysis is specified. Z Transform Basics with Z transform formulas are explained below,

Coding for sources with memory: A drawback of Huffman’s code is that it requires simple probabilities. For real time applications, Huffman encoding becomes impractical as a source statistics are not always known apriori. A code that might be more efficient to use statistical interdependence of the letters in the alphabet along with their individual probabilities of occurrences is the Lempel-ziv algorithm. This algorithm belongs to the class of universal source coding algorithms.The logic behind Lempel-ziv universal coding is as follows:* The compression of an arbitrary binary sequence is possible by coding a series of zeros and ones as some previous such string (prefix string) + one new bit.* The new string formed by such parsing becomes a potential prefix, string for future strings. These variables are called phrases (sub sequences). The phrases are listed in a dictionary or code book which stores the existing phrases and their locations. In encoding a new phrase be specified the location of the existing phrase in the code book and append the new letter.Consider an example:1. Determine the Lempel-ziv code for the given sequence:000101110010100101………….. ?Answer: The given sequence is: 00, 01, 011, 10, 010, 100, 101Depending on this a table can be formed as :

Numeric position

1

2

3

4

5

6

7

8

9

Sub sequence

0

1

00

01

011

10

010

100

101

Numerical representation

11

12

42

21

41

61

62

Binary coding sequence

0010

0011

1001

0100

1000

1100

1101

Maximum number in numerical representation = 66=110No of bits = 3i.e,.0 = 0001 = 0012 = 01014 = 1005 = 1016 = 110The last symbol of each sub sequence in the code book is an innovation sequence corresponding the last bit of each uniform blocks of bits in the binary encoded representation of the data stream represents innovation symbol for the particular subsequence under consideration. The remaining bits provide the equal binary representation of the pointer in the root subsequence that matches the one in question except for the innovation number